metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C10)⋊3D8, C4⋊D4⋊3D5, C5⋊2C8⋊22D4, C5⋊3(C8⋊7D4), C4⋊C4.58D10, C10.55(C2×D8), C4.170(D4×D5), C20⋊7D4⋊23C2, C22⋊1(D4⋊D5), (C2×D4).38D10, (C2×C20).263D4, C20.147(C2×D4), D20⋊6C4⋊35C2, C10.97(C4○D8), D4⋊Dic5⋊15C2, C10.D8⋊36C2, (C22×C10).84D4, C20.183(C4○D4), C4.59(D4⋊2D5), C10.93(C4⋊D4), (C2×C20).357C23, (D4×C10).54C22, (C22×C4).340D10, C23.39(C5⋊D4), (C2×D20).101C22, C4⋊Dic5.142C22, C2.14(Dic5⋊D4), C2.16(D4.8D10), (C22×C20).161C22, (C2×D4⋊D5)⋊10C2, (C5×C4⋊D4)⋊3C2, C2.10(C2×D4⋊D5), (C22×C5⋊2C8)⋊3C2, (C2×C10).488(C2×D4), (C2×C4).105(C5⋊D4), (C5×C4⋊C4).105C22, (C2×C4).457(C22×D5), C22.163(C2×C5⋊D4), (C2×C5⋊2C8).256C22, SmallGroup(320,665)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C10)⋊D8
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, dad=ab5, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 550 in 134 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C5⋊2C8, C5⋊2C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C8⋊7D4, C2×C5⋊2C8, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C5×C22⋊C4, C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C10.D8, D20⋊6C4, D4⋊Dic5, C22×C5⋊2C8, C20⋊7D4, C2×D4⋊D5, C5×C4⋊D4, (C2×C10)⋊D8
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C4○D8, C5⋊D4, C22×D5, C8⋊7D4, D4⋊D5, D4×D5, D4⋊2D5, C2×C5⋊D4, C2×D4⋊D5, Dic5⋊D4, D4.8D10, (C2×C10)⋊D8
►On 160 points
Generators in S160
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 60 40 90 30 70 50 80)(2 59 31 89 21 69 41 79)(3 58 32 88 22 68 42 78)(4 57 33 87 23 67 43 77)(5 56 34 86 24 66 44 76)(6 55 35 85 25 65 45 75)(7 54 36 84 26 64 46 74)(8 53 37 83 27 63 47 73)(9 52 38 82 28 62 48 72)(10 51 39 81 29 61 49 71)(11 105 145 125 155 95 135 115)(12 104 146 124 156 94 136 114)(13 103 147 123 157 93 137 113)(14 102 148 122 158 92 138 112)(15 101 149 121 159 91 139 111)(16 110 150 130 160 100 140 120)(17 109 141 129 151 99 131 119)(18 108 142 128 152 98 132 118)(19 107 143 127 153 97 133 117)(20 106 144 126 154 96 134 116)
(2 10)(3 9)(4 8)(5 7)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 144)(18 143)(19 142)(20 141)(21 29)(22 28)(23 27)(24 26)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 50)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 90)(91 94)(92 93)(95 100)(96 99)(97 98)(101 104)(102 103)(105 110)(106 109)(107 108)(111 124)(112 123)(113 122)(114 121)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(131 154)(132 153)(133 152)(134 151)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)
G:=sub<Sym(160)| (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,60,40,90,30,70,50,80)(2,59,31,89,21,69,41,79)(3,58,32,88,22,68,42,78)(4,57,33,87,23,67,43,77)(5,56,34,86,24,66,44,76)(6,55,35,85,25,65,45,75)(7,54,36,84,26,64,46,74)(8,53,37,83,27,63,47,73)(9,52,38,82,28,62,48,72)(10,51,39,81,29,61,49,71)(11,105,145,125,155,95,135,115)(12,104,146,124,156,94,136,114)(13,103,147,123,157,93,137,113)(14,102,148,122,158,92,138,112)(15,101,149,121,159,91,139,111)(16,110,150,130,160,100,140,120)(17,109,141,129,151,99,131,119)(18,108,142,128,152,98,132,118)(19,107,143,127,153,97,133,117)(20,106,144,126,154,96,134,116), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,124)(112,123)(113,122)(114,121)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(131,154)(132,153)(133,152)(134,151)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)>;
G:=Group( (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,60,40,90,30,70,50,80)(2,59,31,89,21,69,41,79)(3,58,32,88,22,68,42,78)(4,57,33,87,23,67,43,77)(5,56,34,86,24,66,44,76)(6,55,35,85,25,65,45,75)(7,54,36,84,26,64,46,74)(8,53,37,83,27,63,47,73)(9,52,38,82,28,62,48,72)(10,51,39,81,29,61,49,71)(11,105,145,125,155,95,135,115)(12,104,146,124,156,94,136,114)(13,103,147,123,157,93,137,113)(14,102,148,122,158,92,138,112)(15,101,149,121,159,91,139,111)(16,110,150,130,160,100,140,120)(17,109,141,129,151,99,131,119)(18,108,142,128,152,98,132,118)(19,107,143,127,153,97,133,117)(20,106,144,126,154,96,134,116), (2,10)(3,9)(4,8)(5,7)(11,150)(12,149)(13,148)(14,147)(15,146)(16,145)(17,144)(18,143)(19,142)(20,141)(21,29)(22,28)(23,27)(24,26)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,50)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,90)(91,94)(92,93)(95,100)(96,99)(97,98)(101,104)(102,103)(105,110)(106,109)(107,108)(111,124)(112,123)(113,122)(114,121)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(131,154)(132,153)(133,152)(134,151)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155) );
G=PermutationGroup([[(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,60,40,90,30,70,50,80),(2,59,31,89,21,69,41,79),(3,58,32,88,22,68,42,78),(4,57,33,87,23,67,43,77),(5,56,34,86,24,66,44,76),(6,55,35,85,25,65,45,75),(7,54,36,84,26,64,46,74),(8,53,37,83,27,63,47,73),(9,52,38,82,28,62,48,72),(10,51,39,81,29,61,49,71),(11,105,145,125,155,95,135,115),(12,104,146,124,156,94,136,114),(13,103,147,123,157,93,137,113),(14,102,148,122,158,92,138,112),(15,101,149,121,159,91,139,111),(16,110,150,130,160,100,140,120),(17,109,141,129,151,99,131,119),(18,108,142,128,152,98,132,118),(19,107,143,127,153,97,133,117),(20,106,144,126,154,96,134,116)], [(2,10),(3,9),(4,8),(5,7),(11,150),(12,149),(13,148),(14,147),(15,146),(16,145),(17,144),(18,143),(19,142),(20,141),(21,29),(22,28),(23,27),(24,26),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,50),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,80),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,90),(91,94),(92,93),(95,100),(96,99),(97,98),(101,104),(102,103),(105,110),(106,109),(107,108),(111,124),(112,123),(113,122),(114,121),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(131,154),(132,153),(133,152),(134,151),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 40 | 2 | 2 | 2 | 2 | 8 | 40 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D8 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4×D5 | D4⋊2D5 | D4⋊D5 | D4.8D10 |
| kernel | (C2×C10)⋊D8 | C10.D8 | D20⋊6C4 | D4⋊Dic5 | C22×C5⋊2C8 | C20⋊7D4 | C2×D4⋊D5 | C5×C4⋊D4 | C5⋊2C8 | C2×C20 | C22×C10 | C4⋊D4 | C20 | C2×C10 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2×C4 | C23 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 |
Matrix representation of (C2×C10)⋊D8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 35 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 12 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 34 | 35 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,7,0,0,0,0,35,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,0,35,0,0,0,0,7,24,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[34,8,0,0,0,0,35,7,0,0,0,0,0,0,1,21,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
(C2×C10)⋊D8 in GAP, Magma, Sage, TeX
(C_2\times C_{10})\rtimes D_8 % in TeX
G:=Group("(C2xC10):D8"); // GroupNames label
G:=SmallGroup(320,665);
// by ID
G=gap.SmallGroup(320,665);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^5,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations